I had the same problem. Make your way to settings>more>application manager>all>contacts storage. Clear cache. I had to be fast between clearing the pop-up and opening up the next app. I found that if I rebooted the problem would appear again. Ultimately had to do a factory reset. Problem was I deleted an stock app ( contacts ) cause I substituted another app for it. Newbie mistake for me.

I have Samsung Galaxy S4.
Here is one way I know to fix this problem without factory reset.
It is to go settings ->application manger -> all-> and clear data on contact and contact storage in app manager. This will fix the problem. But... Your data for contacts will be wiped away. So remember to BACKUP!
And with this 1 s error thing: you can press ok and go to settings, then press ok when it pops up again and click application manager and etc.. one finger on ok button, and another on one of the following steps. One step at the time.

Another way is factory reset.

SOLUTION FOR THOSE WHO HAD PROBLEMS AFTER ALL THIS (I was one of them):
DO NOT MOVE SKYPE TO SD CARD. LEAVE IT ON THE PHONE MEMORY. It interferes somehow with these contacts things.
THIS WORKED FOR ME!
P.S. I have even tried facory reset and again had "android.process.acore has stopped" problem. And then I have done this with skype, and now everything is just fine.

This worked great for me...the second time I tried it. I had to remove the SD card and the SIM card and then reboot my phone, clear the caches and then put everything back in and reboot it again. I still don't know what caused the error in the first place...but thanks for the tip.

Settings, application manager, then push android menu button on lower left of your phone or tablet, select "Reset app preferences"
This worked pefectly for my Samsung s4. You just have to have quick fingers.

I had this problem. It was caused by me using a phone booster app to disable the Samsung Calendar app.
If it is caused by disabling an app, the quick and easy way to fix is to do the following in between the message popping up (one touch or swipe at a time!):
Swipe the notification bar down and touch the settings icon.
Touch "More".
Touch "Application Manager".
Swipe to the far right "Disabled" list.
Select the suspect app (likely to be Calendar or Contacts, but could be anything!).
Touch "Enable".

If it has worked, the message should stop popping up right away.

I registered on this site just to share this, so I hope it helps somebody!

Latest iteration of this caused by downloading the new Outlook app that replaces Outlook.com. Does the same thing, corrupts the contacts. Apparently you can't have both apps installed at the same time. Above instructions all apply.
Thank you all very much!